LinAlg4.1.10

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If the position vectors of A and B are {\bar  {i}}+3{\bar  {j}}-7{\bar  {k}},5{\bar  {i}}-2{\bar  {j}}+4{\bar  {k}}\, respectively,determine the direction cosines of {\bar  {AB}}\,

Given the position vectors {\bar  {i}}+3{\bar  {j}}-7{\bar  {k}},5{\bar  {i}}-2{\bar  {j}}+4{\bar  {k}}\,

Then {\bar  {AB}}=(5{\bar  {i}}-2{\bar  {j}}+4{\bar  {k}})-({\bar  {i}}+3{\bar  {j}}-7{\bar  {k}})=4{\bar  {i}}-5{\bar  {j}}+11{\bar  {k}}\,

Also |AB|={\sqrt  {16+25+121}}={\sqrt  {162}}=9{\sqrt  {2}}\,

Now unit vector of AB is {\frac  {4}{9{\sqrt  {2}}}}{\bar  {i}}+{\frac  {-5}{9{\sqrt  {2}}}}{\bar  {j}}+{\frac  {11}{9{\sqrt  {2}}}}{\bar  {k}}\,

Therefore,the direction cosines are {\frac  {4}{9{\sqrt  {2}}}},{\frac  {-5}{9{\sqrt  {2}}}},{\frac  {11}{9{\sqrt  {2}}}}\,


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